GLn(Fq)-analogues of factorization problems in the symmetric group

We consider GLn(Fq)-analogues of certain factorization problems in the symmetric group SnSn: rather than counting factorizations of the long cycle (1,2,…,n)(1,2,…,n) given the number of cycles of each factor, we count factorizations of a regular elliptic element given the fixed space dimension of each factor. We show that, as in SnSn, the generating function counting these factorizations has attractive coefficients after an appropriate change of basis. Our work generalizes several recent results on factorizations in GLn(Fq) and also uses a character-based approach.As an application of our results, we compute the asymptotic growth rate of the number of factorizations of fixed genus of a regular elliptic element in GLn(Fq) into two factors as n→∞n→∞. We end with a number of open questions.